Finite element approximation of -surfaces

In this paper a piecewise linear finite element approximation of -surfaces, or surfaces with constant mean curvature, spanned by a given Jordan curve in is considered. It is proved that the finite element -surfaces converge to the exact -surfaces under the condition that the Jordan curve is rectifiable. Several numerical examples are given.

     

Nontrivial Galois module structure of cyclotomic fields

We say a tame Galois field extension with Galois group has trivial Galois module structure if the rings of integers have the property that is a free -module. The work of Greither, Replogle, Rubin, and Srivastav shows that for each algebraic number field other than the rational numbers there will exist infinitely many primes so that for each there is a tame Galois field extension of degree so that has nontrivial Galois module structure. However, the proof does not directly yield specific primes for a given algebraic number field For any cyclotomic field we find an explicit so that there is a tame degree extension with nontrivial Galois module structure.

     

Lower bounds for the total stopping time of iterates

The total stopping time of a positive integer is the minimal number of iterates of the function needed to reach the value , and is if no iterate of reaches . It is shown that there are infinitely many positive integers having a finite total stopping time such that 6.14316 \log n.$"> The proof involves a search of trees to depth 60, A heuristic argument suggests that for any constant , a search of all trees to sufficient depth could produce a proof that there are infinitely many such that \gamma\log n.$">It would require a very large computation to search trees to a sufficient depth to produce a proof that the expected behavior of a ``random' iterate, which is occurs infinitely often.

     

Linear quintuple-product identities

In the first part of this paper, series and product representations of four single-variable triple products , , , and four single-variable quintuple products , , , are defined. Reduced forms and reduction formulas for these eight functions are given, along with formulas which connect them. The second part of the paper contains a systematic computer search for linear trinomial identities. The complete set of such families is found to consist of two 2-parameter families, which are proved using the formulas in the first part of the paper.

     

Newman polynomials with prescribed vanishing and integer sets with distinct subset sums

We study the problem of determining the minimal degree of a polynomial that has all coefficients in and a zero of multiplicity at . We show that a greedy solution is optimal precisely when , mirroring a result of Boyd on polynomials with coefficients. We then examine polynomials of the form , where is a set of positive odd integers with distinct subset sums, and we develop algorithms to determine the minimal degree of such a polynomial. We determine that satisfies inequalities of the form . Last, we consider the related problem of finding a set of positive integers with distinct subset sums and minimal largest element and show that the Conway-Guy sequence yields the optimal solution for , extending some computations of Lunnon.

     

Solving norm equations in relative number fields using -units

In this paper, we are interested in solving the so-called norm equation , where is a given arbitrary extension of number fields and a given algebraic number of . By considering -units and relative class groups, we show that if there exists at least one solution (in , but not necessarily in ), then there exists a solution for which we can describe precisely its prime ideal factorization. In fact, we prove that under some explicit conditions, the -units that are norms are norms of -units. This allows us to limit the search for rational solutions to a finite number of tests, and we give the corresponding algorithm. When is an algebraic integer, we also study the existence of an integral solution, and we can adapt the algorithm to this case.

     

Enhanced accuracy by post-processing for finite element methods for hyperbolic equations

We consider the enhancement of accuracy, by means of a simple post-processing technique, for finite element approximations to transient hyperbolic equations. The post-processing is a convolution with a kernel whose support has measure of order one in the case of arbitrary unstructured meshes; if the mesh is locally translation invariant, the support of the kernel is a cube whose edges are of size of the order of only. For example, when polynomials of degree are used in the discontinuous Galerkin (DG) method, and the exact solution is globally smooth, the DG method is of order in the -norm, whereas the post-processed approximation is of order ; if the exact solution is in only, in which case no order of convergence is available for the DG method, the post-processed approximation converges with order in , where is a subdomain over which the exact solution is smooth. Numerical results displaying the sharpness of the estimates are presented.

     

About the sharpness of the stability estimates in the Kreiss matrix theorem

One of the conditions in the Kreiss matrix theorem involves the resolvent of the matrices under consideration. This so-called resolvent condition is known to imply, for all , the upper bounds and . Here is the spectral norm, is the constant occurring in the resolvent condition, and the order of is equal to .

It is a long-standing problem whether these upper bounds can be sharpened, for all fixed 1$">, to bounds in which the right-hand members grow much slower than linearly with and with , respectively. In this paper it is shown that such a sharpening is impossible. The following result is proved: for each 0$">, there are fixed values 0, K>1$"> and a sequence of matrices , satisfying the resolvent condition, such that for .

The result proved in this paper is also relevant to matrices whose -pseudospectra lie at a distance not exceeding from the unit disk for all 0$">.

     

Maximum norm stability of difference schemes for parabolic equations on overset nonmatching space-time grids

In this paper, theoretical results are described on the maximum norm stability and accuracy of finite difference discretizations of parabolic equations on overset nonmatching space-time grids. We consider parabolic equations containing a linear reaction term on a space-time domain which is decomposed into an overlapping collection of cylindrical subregions of the form , for . Each of the space-time domains are assumed to be independently grided (in parallel) according to the local geometry and space-time regularity of the solution, yielding space-time grids with mesh parameters and . In particular, the different space-time grids need not match on the regions of overlap, and the time steps can differ from one grid to the next. We discretize the parabolic equation on each local grid by employing an explicit or implicit -scheme in time and a finite difference scheme in space satisfying a discrete maximum principle. The local discretizations are coupled together, without the use of Lagrange multipliers, by requiring the boundary values on each space-time grid to match a suitable interpolation of the solution on adjacent grids. The resulting global discretization yields a large system of coupled equations which can be solved by a parallel Schwarz iterative procedure requiring some communication between adjacent subregions. Our analysis employs a contraction mapping argument.

Applications of the results are briefly indicated for reaction-diffusion equations with contractive terms and heterogeneous hyperbolic-parabolic approximations of parabolic equations.

     

A fast algorithm for testing reducibility of trinomials mod 2 and some new primitive trinomials of degree 3021377

The standard algorithm for testing reducibility of a trinomial of prime degree over requires bits of memory. We describe a new algorithm which requires only bits of memory and significantly fewer memory references and bit-operations than the standard algorithm.

If is a Mersenne prime, then an irreducible trinomial of degree is necessarily primitive. We give primitive trinomials for the Mersenne exponents , , and . The results for extend and correct some computations of Kumada et al. The two results for are primitive trinomials of the highest known degree.

     

Three- and four-dimensional -optimal lattice rules of moderate trigonometric degree

A systematic search for optimal lattice rules of specified trigonometric degree over the hypercube has been undertaken. The search is restricted to a population of lattice rules . This includes those where the dual lattice may be generated by points for each of which . The underlying theory, which suggests that such a restriction might be helpful, is presented. The general character of the search is described, and, for , and , , a list of -optimal rules is given. It is not known whether these are also optimal rules in the general sense; this matter is discussed.

     

Strong tractability of multivariate integration using quasi-Monte Carlo algorithms

We study quasi-Monte Carlo algorithms based on low discrepancy sequences for multivariate integration. We consider the problem of how the minimal number of function evaluations needed to reduce the worst-case error from its initial error by a factor of depends on and the dimension . Strong tractability means that it does not depend on and is bounded by a polynomial in . The least possible value of the power of is called the -exponent of strong tractability. Sloan and Wozniakowski established a necessary and sufficient condition of strong tractability in weighted Sobolev spaces, and showed that the -exponent of strong tractability is between 1 and 2. However, their proof is not constructive.

In this paper we prove in a constructive way that multivariate integration in some weighted Sobolev spaces is strongly tractable with -exponent equal to 1, which is the best possible value under a stronger assumption than Sloan and Wozniakowski's assumption. We show that quasi-Monte Carlo algorithms using Niederreiter's -sequences and Sobol sequences achieve the optimal convergence order for any 0$"> independent of the dimension with a worst case deterministic guarantee (where is the number of function evaluations). This implies that strong tractability with the best -exponent can be achieved in appropriate weighted Sobolev spaces by using Niederreiter's -sequences and Sobol sequences.

     

Constructing representations of higher degrees of finite simple groups and covers

Let be a finite group and an irreducible character of . A simple method for constructing a representation affording can be used whenever has a subgroup such that has a linear constituent with multiplicity 1. In this paper we show that (with a few exceptions) if is a simple group or a covering group of a simple group and is an irreducible character of of degree between 32 and 100, then such a subgroup exists.

     

Stability of Runge-Kutta methods for quasilinear parabolic problems

We consider a quasilinear parabolic problem

where , , is a family of sectorial operators in a Banach space with fixed domain . This problem is discretized in time by means of a strongly A()-stable, , Runge-Kutta method. We prove that the resulting discretization is stable, under some natural assumptions on the dependence of with respect to . Our results are useful for studying in norms, , many problems arising in applications. Some auxiliary results for time-dependent parabolic problems are also provided.

     

Upper bounds for the prime divisors of Wendt's determinant

Let be an even integer, . The resultant of the polynomials and is known as Wendt's determinant of order . We prove that among the prime divisors of only those which divide or can be larger than , where and is the th Lucas number, except when and . Using this estimate we derive criteria for the nonsolvability of Fermat's congruence.

     

Sums of heights of algebraic numbers

For , we consider the set . The polynomials are in , with only mild restrictions, and is the Weil height of . We show that this set is dense in for some effectively computable limit point .

     

Asymptotics of recurrence coefficients for orthonormal polynomials on the line--Magnus's method revisited

We use Freud equations to obtain the main term in the asymptotic expansion of the recurrence coefficients associated with orthonormal polynomials for weights on the real line where is an even polynomial of fixed degree with nonnegative coefficients or where . Here for some real -1$">.

     

Weak coupling of solutions of first-order least-squares method

A theoretical analysis of a first-order least-squares finite element method for second-order self-adjoint elliptic problems is presented. We investigate the coupling effect of the approximate solutions for the primary function and for the flux . We prove that the accuracy of the approximate solution for the primary function is weakly affected by the flux . That is, the bound for is dependent on , but only through the best approximation for multiplied by a factor of meshsize . Similarly, we provide that the bound for is dependent on , but only through the best approximation for multiplied by a factor of the meshsize . This weak coupling is not true for the non-selfadjoint case. We provide the numerical experiment supporting the theorems in this paper.

     

Approximation by quadrilateral finite elements

We consider the approximation properties of finite element spaces on quadrilateral meshes. The finite element spaces are constructed starting with a given finite dimensional space of functions on a square reference element, which is then transformed to a space of functions on each convex quadrilateral element via a bilinear isomorphism of the square onto the element. It is known that for affine isomorphisms, a necessary and sufficient condition for approximation of order in and order in is that the given space of functions on the reference element contain all polynomial functions of total degree at most . In the case of bilinear isomorphisms, it is known that the same estimates hold if the function space contains all polynomial functions of separate degree . We show, by means of a counterexample, that this latter condition is also necessary. As applications, we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for serendipity finite elements and for various mixed and nonconforming finite elements.

     

On the modular curves

Let denote an elliptic curve over and the modular curve classifying the elliptic curves over such that the representations of in the 7-torsion points of and of are symplectically isomorphic. In case is given by a Weierstraß equation such that the invariant is a square, we exhibit here nontrivial points of . From this we deduce an infinite family of curves for which has at least four nontrivial points.